Covered in lectures. Check back once the chapter is concluded.
7 Cauchy’s theorem
Definition 7.1 Let \(D\subset\C\) and \(\ga_0, \ga_1\colon[a,b]\to D\) curves in \(D.\)
A homotopy in \(D\) between paths \(\ga_0, \ga_1\) is a continuous map \[\Ga\colon[0,1]\t[a,b]\longra D, (s,t)\longmapsto\Ga_s(t),\] such that \(\Ga_0(t)=\ga_0(t)\) and \(\Ga_1(t)=\ga_1(t)\) for all \(t\in[a,b].\) Then \(\ga_0, \ga_1\) are called (freely) homotopic paths in \(D.\)
If, additionally, \(\Ga_s(a)=p\) and \(\Ga_s(b)=q\) are constant in \(s\in[0,1],\) we call \(\Ga\) a path homotopy in \(D\) and \(\ga_0, \ga_1\) path-homotopic in \(D\).
If \(\ga_0, \ga_1\) are loops, a homotopy of loops in \(D\) is a homotopy \(\Ga\) in \(D\) with the additional property that \(\Ga_s\) is a loop for each \(s\in[0,1].\) Then \(\ga_0, \ga_1\) are called (freely) homotopic loops in \(D.\)
A loop is null-homotopic in \(D\) if there is a homotopy of loops in \(D\) to the constant loop.
Remark 7.1. In the following we will also suppose that \(\Ga\) is piecewise C1, meaning there exist subdivisions \[0=s_0<s_1<\cdots<s_m=1,\qquad a=t_0<t_1<\cdots<t_n=b\]
such that each restriction \(\Ga|_{[s_{j-1},s_j]\t[t_{k-1},t_k]}\) is continuously differentiable.
Example 7.1
Covered in lectures. Check back once the chapter is concluded.
Theorem 7.1 (Cauchy’s theorem) Let \(f\colon U\to \C\) be a holomorphic function on an open set. Let \(\ga_0, \ga_1\) be piecewise C1 curves in \(U\) that are path-homotopic in \(U.\) Then
\[\int_{\ga_0}f(z)dz = \int_{\ga_1}f(z)dz.\]
Proof.
Covered in lectures. Check back once the chapter is concluded.
Theorem 7.2 Let \(\ga_0, \ga_1\) be piecewise C1 loops in \(U\) that are freely homotopic in \(U.\) Suppose \(f\colon U\to\C\) is a holomorphic function. Then
\[\int_{\ga_0}f(z)dz = \int_{\ga_1}f(z)dz.\]In particular, if \(\ga\) is a loop that is homotopic in \(U\) to the constant loop, then\[\int_\ga f(z)dz=0. \tag{7.1}\]
Proof.
Covered in lectures. Check back once the chapter is concluded.
Questions for further discussion
Covered in lectures. Check back once the chapter is concluded.
Give a counterexample to Cauchy’s theorem when (i) \(\ga\) is not null-homotopic (ii) \(f(z)\) is not holomorphic
Does Cauchy’s theorem hold for \(f(z)=\ol{z}\)?
7.1 Exercises
Sketch the curves \[\begin{align*} \ga_0\colon[-1,1]&\longra\C,&\ga_0(t)&=t,\\ \ga_1\colon[-1,1]&\longra\C,&\ga_1(t)&=e^{i\pi\frac{1-t}{2}} \end{align*}\] and show that they are path-homotopic.
- Let \(0<r_0<r_1\) and \(z_0\in\C.\) Prove that the loops \(\partial D_{r_0}(z_0)\) and \(\partial D_{r_1}(z_0)\) are freely homotopic in the closed annulus \(\ol A_{r_0,r_1}(z_0).\)
Let \(\al+i\be\in\C.\) Determine \[\int_a^b e^{(\al+i\be)t} dt\] to compute \[\int_a^b e^{\al t}\cos(\be t)dt.\]
Let \(\C^-=\C\setminus(-\iy,0]\) be the slit plane.
- Show that any two points in \(\C^-\) may be connected by a path in \(\C^-.\) Hence \(\C^-\) is path-connected.
- Show that every closed curve \(\ga\colon[a,b]\to\C^-\) is null-homotopic in \(\C^-\) by finding a homotopy \[\Ga\colon[0,1]\t[a,b]\longra\C^-, (s,t)\longmapsto\Ga_s(t)\] satisfying \(\Ga_0(t)=\ga(t)\) and \(\Ga_1(t)=1\) for all \(t\in[a,b].\) Hence \(\C^-\) is simply-connected.
- Prove the analogues of a. and b. for a disk \(D_r(z_0).\)
- Use Cauchy’s Theorem to prove that the punctured plane \(\C^\t=\C\setminus\{0\}\) is not simply-connected. That is, there exists a closed curve in \(\C^\t\) that is not null-homotopic in \(\C^\t.\)
Let \(0<b<1.\)
- Using the geometric series, find the power series expansion \[\frac{1}{z-1/b}=\sum_{n=0}^\iy a_n(z-b)^n\] with center \(z_0=b\) and determine the radius of convergence \(\rho.\)
- Use a. to show that for all \(0<r<\rho\) \[\int_{\partial D_r(b)}\frac{dz}{(z-b)(z-1/b)}=\frac{2\pi i}{b-1/b}.\]
- Use c. to compute \[\int_0^{2\pi}\frac{dt}{1-2b\cos(t)+b^2}.\]
Let \(\ga\colon[0,1]\to D\) be a curve in \(D\subset\C\) and let \(-\ga\colon[0,1]\to D,\) \((-\ga)(t)=\ga(1-t)\) be the opposite curve. Prove that \(\ga\ast(-\ga)\) is path-homotopic to a constant loop.